2. A relation is a set of ordered pairs.
The domain is the set of all x values in the relation
domain = {-1,0,2,4,9}
These are the x values written in a set from smallest to largest
{(2,3), (-1,5), (4,-2), (9,9), (0,-6)}
These are the y values written in a set from smallest to largest
range = {-6,-2,3,5,9}
The range is the set of all y values in the relation
This is a
relation
3. A relation assigns the x’s with y’s
1
2
3
2
4
5
6
8
10
Domain (set of all x’s)
Range (set of all y’s)
4
This relation can be written {(1,6), (2,2), (3,4), (4,8), (5,10)}
4. A function ff from set A to set B is a rule of correspondence
A function from set A to set B is a rule of correspondence
that assigns to each element x in the set A exactly one
that assigns to each element x in the set A exactly
element y in the set B.
element y in the set B.
are
x’s ned
All sig
as
1
2
3
4
5
Set A is the domain
2
4
6
8
10
No
xh
tha as
m
ass n one ore
ig n y
ed
Set B is the range
This is a function
Must use all the x’s
---it meets our
conditions
The x value can only be assigned to one y
5. Let’s look at another relation and decide if it is a function.
The second condition says each x can have only one y, but it CAN
be the same y as another x gets assigned to.
are
x’s ned
All sig
as
1
2
3
4
5
Set A is the domain
2
4
6
8
10
No
xh
tha as
m
ass n one ore
ig n y
ed
Set B is the range
This is a function
Must use all the x’s
---it meets our
conditions
The x value can only be assigned to one y
6. good example that you can “relate” to is students in our
AAgood example that you can “relate” to is students in our
maths class this semester are set A. The grade they earn out
maths class this semester are set A. The grade they earn out
of the class is set B. Each student must be assigned grade
of the class is set B. Each student must be assigned aagrade
and can only be assigned ONE grade, but more than one
and can only be assigned ONE grade, but more than one
student can get the same grade (we hope so---we want lots
student can get the same grade (we hope so---we want lots of
of A’s). example shown on the previous screen had each
A’s). TheThe example show on the previous screen had each
student getting the same grade. That’s okay.
student getting the same grade. That’s okay.
1
2
3
4
5
2
4
6
8
10
2 was assigned both 4 and 10
Is the relation shown above a function?
NO
Why not???
7. Check this relation out to determine if it is a function.
It is not---3 didn’t get assigned to anything
Comparing to our example, a student in maths must receive a grade
1
2
3
4
5
Set A is the domain
2
4
6
8
10
Set B is the range
This is not a
Must use all the x’s
function---it
doesn’t assign
each x with a y The x value can only be assigned to one y
8. Check this relation out to determine if it is a function.
This is fine—each student gets only one grade. More than one can
get an A and I don’t have to give any D’s (so all y’s don’t need to be
used).
1
2
3
4
5
2
4
6
8
10
Set A is the domain
This is a function
Set B is the range
Must use all the x’s
The x value can only be assigned to one y
9. We commonly call functions by letters. Because function
starts with f, it is a commonly used letter to refer to
functions.
f ( x ) = 2 x − 3x + 6
2
This means
the right
hand side is
a function
called f
This means
the right hand
side has the
variable x in it
The left side DOES NOT
MEAN f times x like
brackets usually do, it
simply tells us what is on
the right hand side.
The left hand side of this equation is the function notation.
It tells us two things. We called the function f and the
variable in the function is x.
10. Remember---this tells you what
is on the right hand side---it is
not something you work. It says
that the right hand side is the
function f and it has x in it.
f ( x ) = 2 x − 3x + 6
2
f ( 2 ) = 2( 2 ) − 3( 2 ) + 6
2
f ( 2 ) = 2( 4 ) − 3( 2 ) + 6 = 8 − 6 + 6 = 8
So we have a function called f that has the variable x in it.
Using function notation we could then ask the following:
This means to find the function f and instead of
having an x in it, put a 2 in it. So let’s take the
Find f (2).
function above and make brackets everywhere
the x was and in its place, put in a 2.
Don’t forget order of operations---powers, then
multiplication, finally addition & subtraction
11. Find f (-2).
f ( x ) = 2 x − 3x + 6
2
f ( − 2 ) = 2( − 2 ) − 3( − 2 ) + 6
2
f ( − 2 ) = 2( 4 ) − 3( − 2 ) + 6 = 8 + 6 + 6 = 20
This means to find the function f and instead of having an x
in it, put a -2 in it. So let’s take the function above and make
brackets everywhere the x was and in its place, put in a -2.
Don’t forget order of operations---powers, then
multiplication, finally addition & subtraction
12. f ( x ) = 2 x − 3x + 6
2
Find f (k).
f ( k ) = 2( k ) − 3( k ) + 6
2
f ( k ) = 2( k ) − 3( k ) + 6 = 2k − 3k + 6
2
2
This means to find the function f and instead of having an x
in it, put a k in it. So let’s take the function above and make
brackets everywhere the x was and in its place, put in a k.
Don’t forget order of operations---powers, then
multiplication, finally addition & subtraction
13. f ( x ) = 2 x − 3x + 6
2
Find f (2k).
f ( 2k ) = 2( 2k ) − 3( 2k ) + 6
2
f ( 2k ) = 2( 4k ) − 3( 2k ) + 6 = 8k − 6k + 6
2
2
This means to find the function f and instead of having an x in
it, put a 2k in it. So let’s take the function above and make
brackets everywhere the x was and in its place, put in a 2k.
Don’t forget order of operations---powers, then
multiplication, finally addition & subtraction
14. Let's try a new function
Find g(1)+ g(-4).
g( x) = x − 2x
2
g (1) = (1) − 2(1) = −1
2
g ( − 4 ) = ( − 4 ) − 2( − 4 ) = 16 + 8 = 24
2
So g (1) + g ( − 4 ) = −1 + 24 = 23
15. The last thing we need to learn about functions for
this section is something about their domain. Recall
domain meant "Set A" which is the set of values you
plug in for x.
For the functions we will be dealing with, there
are two "illegals":
1. You can't divide by zero (denominator (bottom)
of a fraction can't be zero)
2. You can't take the square root (or even root) of
a negative number
When you are asked to find the domain of a function,
you can use any value for x as long as the value
won't create an "illegal" situation.
16. Find the domain for the following functions:
Since no matter what value you
choose for x, you won't be dividing
f x = 2 x − 1 by zero or square rooting a negative
number, you can use anything you
Note: There is
want so we say the answer is:
nothing wrong with
the top = 0 just means All real numbers x.
( )
the fraction = 0
x+3
g ( x) =
x−2
illegal if this
is zero
If you choose x = 2, the denominator
will be 2 – 2 = 0 which is illegal
because you can't divide by zero.
The answer then is:
All real numbers x such that x ≠ 2.
means does not equal
17. Let's find the domain of another one:
h( x ) = x − 4
Can't be negative so must be ≥ 0
x−4≥0
solve
this
x≥4
We have to be careful what x's we use so that the second
"illegal" of square rooting a negative doesn't happen. This
means the "stuff" under the square root must be greater
than or equal to zero (maths way of saying "not negative").
18. Summary of How to Find the
Domain of a Function
• Look for any fractions or square roots that could cause one
of the two "illegals" to happen. If there aren't any, then the
domain is All real numbers x.
• If there are fractions, figure out what values would make the
bottom equal zero and those are the values you can't use.
The answer would be: All real numbers x such that x ≠
those values.
• If there is a square root, the "stuff" under the square root
cannot be negative so set the stuff ≥ 0 and solve, whatever
you got when you solved.
NOTE: Of course your variable doesn't have to be x, can be
whatever is in the problem.
